To provide an example of the results obtained,In this work we are only concerned with the characterizations of capacities expressed in potential theoretical terms, leaving aside those given in terms of transfinite diameters, Chebyshev constants, extremal lengths, or Dirichlet integrals of suitable functions (cf. e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}).
consider a perfect kernel \(\kappa \)  satisfying the first and second maximum principles (for definitions see Sect. ), and let \(\mathcal {E}^+\)  stand for the cone of all positive (scalar) Radon measures \(\mu \)  on \(X\)  of finite energy:
\(\kappa (\mu ,\mu ):=\int \kappa (x,y)\,d(\mu \otimes \mu )(x,y)=\int \kappa \mu \,d\mu <\infty ,\) 
